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AGL(1,𝔽11)×C3⋊(D19⋊C3) #49178509

order37620
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productAGL(1,𝔽11)×C3⋊(D19⋊C3)
unambiguous semidirect productD11⋊(C5×C3⋊(D19⋊C3))
unambiguous semidirect product(C3×AGL(1,𝔽11))⋊(D19⋊C3)
unambiguous semidirect product(C19⋊C3×AGL(1,𝔽11))⋊D3
unambiguous semidirect product(D11×C3⋊(D19⋊C3))⋊C5
unambiguous semidirect product(AGL(1,𝔽11)×D57)⋊C3

Splittings

Fitting, socle, derivedC3×C11×C19 ⋊ C22×C3×C5
HallC3×C11⋊C5×C19⋊C3 ⋊ C22
HallD11×C3⋊(D19⋊C3) ⋊ C5
HallC3×C11×C19⋊C3 ⋊ C22×C5
HallC19×C11⋊C5 ⋊ C2×C3×D3
HallC11×C19 ⋊ C2×C3×C5×D3
HallC3×C19⋊C3 ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ C2×C3⋊(D19⋊C3)
SylowC19 ⋊ C3×D3×AGL(1,𝔽11)
SylowC11 ⋊ C2×C5×C3⋊(D19⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C11×C19C22×C3×C5
LCSG ▷ C3×C11×C19
UCSC1

176 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 627

A minimal faithful permutation representation — 2 generators on 33 points:

(2,3)(4,5,10,6,12)(7,14,13,8,11)(16,20,21,17,18,22)(19,28,25,23,24,30)(26,31,33,32,29,27);(1,2,3)(5,6)(7,9)(8,11)(10,14)(12,13)(15,16,19,25,31,28,32,18,22,27,33,21,20,26,24,29,30,23,17)

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37620, 218);